L(s) = 1 | − 2-s + 4-s + 4·5-s − 7-s − 8-s − 4·10-s + 6·11-s − 4·13-s + 14-s + 16-s + 4·17-s − 19-s + 4·20-s − 6·22-s − 2·23-s + 11·25-s + 4·26-s − 28-s − 2·29-s + 4·31-s − 32-s − 4·34-s − 4·35-s + 2·37-s + 38-s − 4·40-s − 6·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.78·5-s − 0.377·7-s − 0.353·8-s − 1.26·10-s + 1.80·11-s − 1.10·13-s + 0.267·14-s + 1/4·16-s + 0.970·17-s − 0.229·19-s + 0.894·20-s − 1.27·22-s − 0.417·23-s + 11/5·25-s + 0.784·26-s − 0.188·28-s − 0.371·29-s + 0.718·31-s − 0.176·32-s − 0.685·34-s − 0.676·35-s + 0.328·37-s + 0.162·38-s − 0.632·40-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.962236938\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.962236938\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + T \) |
good | 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.170766396680054577994306068589, −8.478463677722098593043536921802, −7.26142226179841058711952374724, −6.66923383553576609964750752019, −5.99160831439827741765439496257, −5.32705438536939705343146893129, −4.07413511690756283882706586153, −2.85126478970855653424985240380, −1.96927041097654768116323530907, −1.07395817415686514733421458233,
1.07395817415686514733421458233, 1.96927041097654768116323530907, 2.85126478970855653424985240380, 4.07413511690756283882706586153, 5.32705438536939705343146893129, 5.99160831439827741765439496257, 6.66923383553576609964750752019, 7.26142226179841058711952374724, 8.478463677722098593043536921802, 9.170766396680054577994306068589